Well, it depends on the depth. I don't think it will be possible without King term if you restrict the search depth to 1 ply, for instance. In situations other than a simple rule-of-squares race, you need more King moves than Pawn moves to win against best defence. E.g. with Ke4, Pe3 vs Ke6 (btm) after Kd6 both Kf5 and Kf4 are winning, but only Kf5 is progress. So if because of move-gen ordering it happens to prefer Kf4, it would never make progress. Relying on rep-draw code to force progress is a tricky thing, because it also often backfires; In the given example you might think that the second time it would play Kf5 because it now recognizes Kf4 as a draw, but in fact there won't be a second time, because after Kd6, Kf4, Ke6 the only winning move Ke4 is now tainted as rep-draw, so it will accept that the position is drawn now, and pick a random drawing move.

Restricting the search to 1 ply might seem a bit extreme, but it is of course exactly what you do if you want to build a recognizer for this end-game.